What is the best way to clean my v pale bathroom floor tiles and grout ?

[Kitsmummy]

We cleverly used pale flooring and grout (stupid stupid stupid) and parts of the grouting (eg most heavy traffic area) gets quite grubby and the tiles don't fully clean with the steamer either.

Can anyone tell me the best method for cleaning this? A steam mop doesn't make the grade, I need a more "scrubby" method than mopping I think and I'm not sure which product to use. Don't really want to use bleach either as the adjoining hall has new carpet and I'm paranoid about spots of bleach getting onto the carpet.

A scrubbing brush and hot, soapy water? Could you replace the grout with dark grey? Yes it will look different, but no more cleaning woes.

My tiles aren't super smooth, so every so often I use bleach and a scrubbing brush (and a toothbrush in the corners). If you don't want to use bleach, something like Flash should do the job I would think.